Isometric paper, 5 mm

What this sheet is

Isometric drawing paper for A4, 210 mm by 297 mm, at a 5 mm pitch. Three families of lines cross the page: one flat family running left to right, and two sloped families set at sixty degrees either side of it. Together they come to 346 ruled lines, and where they cross they leave a field of small triangles rather than squares.

The sheet is laid out on a 5 mm module, thirty-eight across and fifty-five down, 2,090 in all, which puts 190 mm of ruled width on the 210 mm sheet and 275 mm of ruled height on the 297 mm one. The ruling is centred in that area: the fraction of a module left over once the whole ones have been counted is halved and split between the two margins, instead of all collecting against one edge.

Why the rows sit at 0.866 of the pitch

The flat lines are not 5 mm apart. They sit at the pitch multiplied by the sine of sixty degrees, about 0.866, which at a 5 mm pitch puts them 4.33 mm apart. That factor is the point of the sheet. It is what makes the triangle formed by two sloped lines and one flat one equilateral, all three sides 5 mm, rather than merely pointy.

Get the factor wrong and the sheet looks convincing and draws badly. Space the rows at a full 5 mm instead of 4.33 and every triangle on the page is taller than it is wide. Nothing about the page announces this, because it still reads as isometric paper at arm’s length, but a cube drawn on it comes out stretched along the vertical and no amount of care with the pencil will straighten it. The three axes stop being interchangeable, so a shape moved from one face of the cube to another changes shape as it goes.

Printing it at true size

Scaling is the thing to watch, and isometric paper hides it better than most rulings do. A print dialogue left on Fit to page scales the sheet uniformly, and a uniform scale leaves every angle exactly where it was: sixty degrees stays sixty degrees, and the triangles stay equilateral. Only the size changes. A4 onto US Letter prints at 94.1%, which takes the 5 mm pitch down to 4.7 mm and the rows from 4.33 mm to about 4.07 mm. Letter onto A4 prints at 97.3%, leaving the pitch at 4.86 mm.

So the sheet will look right and measure wrong, which matters the moment a drawing has to be scaled off it, or two sheets have to be laid over one another. Set the dialogue to 100%, or Actual size, with page scaling off. Then measure the bar printed in the foot margin: its length is stated beside it, and if a ruler agrees with the stated figure, the pitch on the page is the pitch that was asked for.

What else the bench will set

The pitch is the first thing to change, coarser for quick sketching and finer where a drawing has to carry detail. The paper can be swapped for A5 at 148 by 210 mm, US Letter at 215.9 by 279.4 or Legal at 215.9 by 355.6, and the sheet set upright or landscape, which suits a long object better than a tall one. The ink can be taken down until the ruling is faint enough to draw straight over and still read through. Every fifth line can be drawn heavier, which gives the eye something to count by on a field this dense, and a margin rule and a name-and-date line can be added at the edge.